scholarly journals Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations

2009 ◽  
Vol 8 (2) ◽  
pp. 683-688
Author(s):  
Yasuhiro Fujita ◽  
◽  
Katsushi Ohmori ◽  
Keyword(s):  
Mather Theory ◽  
Jacobi Equations

scholarly journals The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures

2020 ◽  
Vol 54 (6) ◽  
pp. 1883-1915
Author(s):  
Diogo A. Gomes ◽  
Xianjin Yang
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Hamiltonian Dynamics ◽  
Mean Field ◽  
Mean Field Games ◽  
Mather Theory ◽  
Jacobi Equations ◽  
Continuous Setting ◽  
Riemannian Flow ◽  
Effective Hamiltonians

Effective Hamiltonians arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry–Mather theory. In Aubry–Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

scholarly journals Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

2011 ◽  
Vol 23 (1) ◽  
pp. 377-394
Author(s):  
R. L. Foote ◽  
C. K. Han ◽  
J. W. Oh
Keyword(s):  
Jacobi Equations

scholarly journals Non-local Discrete $$\infty $$ ∞ -Poisson and Hamilton Jacobi Equations

2015 ◽  
Vol 55 (2) ◽  
pp. 229-241 ◽  
Author(s):  
Matthieu Toutain ◽  
Abderrahim Elmoataz ◽  
François Lozes ◽  
Alamin Mansouri
Keyword(s):  
Non Local ◽  
Jacobi Equations
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